Ok...another "how do I get the most accuracy out of my _____________ semi-auto" thread, huh?

Here we go again...
As previously mentioned, a trigger (whether you work over the LMT or purchase a suitable replacement...like a Geissele for example) will help to improve things somewhat (at least in comparison with the stock factory triggers). How much will depend greatly on how bad the current trigger really is, but I generally see improvement for most shooters with quality triggers versus stock, mil-spec units.
Additionally...you didn't mention what form of sights/optic/mount/etc. you are running, but I see a lot of room for improvement on some setups there (again...depending heavily on how you have it squared away now). Plenty of people will never fully realize the potential of a particular weapon system because they buy an expensive rifle and then equip it with sub-standard optics, rings, mounts, etc. and then don't install them correctly. Again...not an accusation...just an observation over MANY years now. Much accuracy can be gained (or lost) from the sighting system you run and how it is installed.
As mentioned above, the rifle's barrel is also a huge factor in accuracy potential. The 20" SS tubes on some of the LMTs are nothing to sneeze at, but some of their other options are...well...not as accurate. There may be the potential to gain additional accuracy from a different barrel, but how much more will greatly depend on several factors.
Also, I see lots of accuracy potential lost or otherwise abandoned due to the use of garbage ammo. You can't feed any rifle surplus ammo or "cheap" so-called match ammo (like PRVI for example) and expect to fulfill the rifle's accuracy potential. Whether quality ammo like FGMM, BH, SWA, etc., etc., or your own reloads tailored to the rifle...you have to pay to play in terms of achieving optimum accuracy!
Finally, never underestimate the deterioration in accuracy potential caused on any platform due to shooter error. Depending on your particular experience with a .308 gasser and how you are operating the rifle, you may or may not be achieving optimum results. Shooting a large-frame gasser is not like shooting a basic AR-15 or anything close to a precision bolt gun. Any lapses in fundamentals, minor or otherwise, from trigger control to recoil mgt to...well, you get the idea...and you can really suffer from accuracy woes having nothing to do with the rifle itself.
From there, you have to determine whether your current level of accuracy (whether with the trigger and/or other upgrades) is or is NOT within the confines of what the platform is capable of achieving. You can't get blood from a turnip and attempting to get 1/2 MOA accuracy from that rifle is going to result in a lot of wasted time and ammo, not to mention cost and frustration!

Knowing more about your setup (hypothetical or otherwise given your description saying "Lets say [you] have a LMT MWSE 20"), as well as the results you are currently achieving and with what ammo, might go a bit further in helping you to get the most out of the rifle.
